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Fractal uncertainty principle with explicit exponent

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Abstract

We prove an explicit formula for the dependence of the exponent \(\beta \) in the fractal uncertainty principle of Bourgain–Dyatlov (Ann Math 187:1–43, 2018) on the dimension \(\delta \) and on the regularity constant \(C_R\) for the regular set. In particular, this implies an explicit essential spectral gap for convex co-compact hyperbolic surfaces when the Hausdorff dimension of the limit set is close to 1.

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References

  1. Beurling, A., Malliavin, P.: On Fourier transforms of measures with compact support. Acta Math. 107, 291–309 (1962)

    Article  MathSciNet  Google Scholar 

  2. Borthwick, D.: Spectral Theory of Infinite-Area Hyperbolic Surface, 2nd edn. Birkhäuser, Boston (2016)

    MATH  Google Scholar 

  3. Bourgain, J.: Private communication

  4. Bourgain, J., Dyatlov, S.: Spectral gaps without the pressure condition. Ann. Math. 187, 1–43 (2018)

    Article  MathSciNet  Google Scholar 

  5. Dyatlov, S., Jin, L.: Resonances for open quantum maps and a fractal uncertainty principle. Commun. Math. Phys. 354, 269–316 (2017)

    Article  MathSciNet  Google Scholar 

  6. Dyatlov, S., Jin, L.: Dolgopyat’s method and the fractal uncertainty principle. Anal. PDE 11, 1457–1485 (2018)

    Article  MathSciNet  Google Scholar 

  7. Dyatlov, S., Jin, L.: Semiclassical measures on hyperbolic surfaces have full support. Acta Math. 220, 297–339 (2018)

    Article  MathSciNet  Google Scholar 

  8. Dyatlov, S., Jin, L., Nonnenmacher, S.: Control of eigenfunctions on surfaces of variable curvature, preprint. arXiv:1906.08923

  9. Dyatlov, S., Zahl, J.: Spectral gaps, additive energy, and a fractal uncertainty principle. Geom. Funct. Anal. 26, 1011–1094 (2016)

    Article  MathSciNet  Google Scholar 

  10. Dyatlov, S., Zworski, M.: Fractal uncertainty for transfer operators. Int. Math. Res. Not. arXiv:1710.05430(to appear)

  11. Havin, V., Jöricke, B.: The Uncertainty Principle in Harmonic Analysis, Ergeb. Math. Grenzgeb. 28. Springer, Berlin (1994)

    Book  Google Scholar 

  12. Han, R., Schlag, W.: A higher dimensional Bourgain-Dyatlov fractal uncertainty principle. Anal. PDE. arXiv:1805.04994(to appear)

  13. Jin, L.: Damped wave equations on compact hyperbolic surfaces, preprint. arXiv:1712.02692

  14. Jin, L.: Control for Schrödinger equation on hyperbolic surfaces. Math. Res. Lett. 25, 1865–1877 (2018)

    Article  MathSciNet  Google Scholar 

  15. Koosis, P.: Leçons sur la théorème de Beurling et Malliavin. Univ. Montréal, Les Publications CRM, Montreal (1996)

    MATH  Google Scholar 

  16. Malliavin, P.: On the multiplier theorem for Fourier transforms of measures with compact support. Ark. Mat. 17, 69–81 (1979)

    Article  MathSciNet  Google Scholar 

  17. Mashreghi, J., Nazarov, F., Havin, V.: Beurling–Malliavin multiplier theorem: the seventh proof. St. Petersb. Math. J. 17, 699–744 (2006)

    Article  MathSciNet  Google Scholar 

  18. Rudnick, Z., Sarnak, P.: The behaviour of eigenstates of arithmetic hyperbolic manifolds. Commun. Math. Phys. 161, 195–213 (1994)

    Article  MathSciNet  Google Scholar 

  19. Zworski, M.: Mathematical study of scattering resonances. Bull. Math. Sci. 7, 1–85 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are very grateful to Semyon Dyatlov for encouragement and helpful discussions and to Kiril Datchev for many useful suggestions. We are especially thankful to Jean Bourgain for sharing with us some preliminary notes [3] about the multiplier theorem and Hörmander’s \({\bar{\partial }}\)-theorem, and to Alexander Sodin for explaining to us the background of the multiplier theorems in the context of complex analysis during the Emerging Topics Working Groups at IAS in fall 2017. We are also indebted to Rui Han and Wilhelm Schlag for pointing out an error in the initial version where we miss an additional exponential in the computation, as well as the anonymous referee for carefully reading the paper and providing many useful suggestions. Finally we would also like to thank BICMR at Peking University for hospitality during our visit in May 2017 where we started the project, Sun Yat-Sen University and YMSC at Tsinghua University where part of the project was finished, and Institute for Advanced Study for hosting a week-long workshop on the topic of quantum chaos and fractal uncertainty principle. RZ is supported by NSF under Grant no. 1638352 and the James D. Wolfensohn Fund. LJ is supported by Recruitment Program of Young Overseas Talent Plan.

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Correspondence to Ruixiang Zhang.

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Communicated by Loukas Grafakos.

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Jin, L., Zhang, R. Fractal uncertainty principle with explicit exponent. Math. Ann. 376, 1031–1057 (2020). https://doi.org/10.1007/s00208-019-01902-8

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  • DOI: https://doi.org/10.1007/s00208-019-01902-8

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